# Galois field
## English
### Etymology

Named after French mathematician Évariste Galois (1811–1832).

### Noun

Galois field (plural Galois fields)
* (algebra) A finite field; a field that contains a finite number of elements.
    The Galois field $\mathrm {GF} (p^{n})$ has order $p^{n}$ and characteristic $p$.
    The Galois field $\mathrm {GF} (p^{n})$ is a finite extension of the Galois field $\mathrm {GF} (p)$ and the degree of the extension is $n$.
    The multiplicative subgroup of a Galois field is cyclic.
    A Galois field $\mathbb {F} _{p^{n}}$ is isomorphic to the quotient of the polynomial ring $\mathbb {F} _{p}$ adjoin $x$ over the ideal generated by a monic irreducible polynomial of degree $n$. Such an ideal is maximal and since a polynomial ring is commutative then the quotient ring must be a field. In symbols: $\mathbb {F} _{p^{n}}\cong {\mathbb {F} _{p}[x] \over ({\hat {f}}_{n}(x))}$.
** 1958 [Chelsea Publishing Company], Hans J. Zassenhaus, The Theory of Groups, 2013, Dover, unnumbered page,
    A field with a finite number of elements is called a Galois field.
    The number of elements of the prime field $k$ contained in a Galois field $K$ is finite, and is therefore a natural prime $p$.
** 2001, Joseph E. Bonin, A Brief Introduction To Matroid Theory‎, retrieved 2016-05-05:The case of most interest to us will be that in which F is a finite field, the Galois field GF(q) for some prime power q. If q is prime, this field is $\mathbb {Z} _{q}$, the integers $0,1,\dots ,q-1$ with arithmetic modulo q.
** 2006, Debojyoti Battacharya, Debdeep Mukhopadhyay, D. RoyChowdhury, A Cellular Automata Based Approach for Generation of Large Primitive Polynomial and Its Application to RS-Coded MPSK Modulation, Samira El Yacoubi, Bastien Chopard, Stefania Bandini (editors), Cellular Automata: 7th International Conference, Proceedings, Springer, LNCS 4173, page 204,
    Generation of large primitive polynomial over a Galois field has been a topic of intense research over the years. The problem of finding a primitive polynomial over a Galois field of a large degree is computationaly expensive and there is no deterministic algorithm for the same.

#### Usage notes

* For a given order, if a Galois field exists, it is unique, up to isomorphism.
* Generally denoted $\mathrm {GF} (n)$ (but sometimes $\mathbb {F} _{n}$), where $n$ is the number of elements, which must be a positive integer power of a prime.
* Although, strictly speaking, the "field of one element" does not exist (it is not a field in classical algebra), it is occasionally discussed in terms of how it might be meaningfully defined. Were it a meaningful concept, it would be a Galois field. It may be denoted $\mathbb {F} _{1}$ or, more jocularly, $\mathbb {F} _{un}$ (pun intended).

#### Hypernyms

* perfect field

### Further reading

* Finite field arithmetic on Wikipedia.Wikipedia
* Finite ring on Wikipedia.Wikipedia
* Finite group on Wikipedia.Wikipedia
* Field with one element on Wikipedia.Wikipedia
* Galois geometry on Wikipedia.Wikipedia
* Galois theory on Wikipedia.Wikipedia
* Hamming space on Wikipedia.Wikipedia
* Galois field on Encyclopedia of Mathematics
* Galois field structure on Encyclopedia of Mathematics
* Finite Field on Wolfram MathWorld